New bounds for a hypergraph Bipartite Turan problem

Let $t$ be an integer such that $tgeq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples ${a,x_i,y_i}$, ${b,x_i,y_i}$ for $1 le i le t$, where the elements $a, b, x_1, x_2, ldots, x_t,$ $y_1, y_2, ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraete, where the special case $t=2$ was a problem of ErdH{o}s that was studied by various authors. Mubayi and Verstraete proved that $ex(n,K_{2,t}^{(3)})<t^4binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})geq frac{2t-1}{3} binom{n}{2}$. These bounds together with a standard argument show that $g(t):=lim_{nto infty} ex(n,K_{2,t}^{(3)})/binom{n}{2}$ exists and that [frac{2t-1}{3}leq g(t)leq t^4.] Addressing the question of Mubayi and Verstraete on the growth rate of $g(t)$, we prove that as $t to infty$, [g(t) = Theta(t^{1+o(1)}).]